A new type of half-metallic fully compensated ferrimagnet

Half-metallic fully compensated ferrimagnets (HM-FCFMs) constitute a special class of half-metals exhibiting zero magnetization at zero temperature. While there have been a number of theoretical studies predicting the existence of such materials over the last 25 years, very few of those have been synthesized and observed that they exhibit expected properties. Herein, we demonstrate that a NiAs-type hexagonal-structured (CrFe)S compound could serve as an HM-FCFM material. It has a half-metallic nature of 100% spin-polarised Fermi surfaces and yet zero magnetisation at the ground state. The magnetisation shows linear behaviour as a function of the magnetic field at temperatures below the compensation temperature (~ 190 K). In addition, it shows a high magnetic coercivity of 3.8 T at 300 K. These magnetic features contribute to a significant development in the application of HM-FCFMs for spintronics devices.


Crystal structure
The powder XRD profile of the sample synthesized by sintering and quenching (Fig. S2A) showed that the sample had a hexagonal NiAs-type structure. To evaluate this result in detail, we simulated the XRD profiles based on possible NiAs-type (CrFe)S structures (Figs. S2(B-E)). Figure S2B shows the XRD profile corresponding to the stoichiometric CdI 2 -type ordered structure illustrated in Fig. 1D, while Fig.  S2C shows the stoichiometric NiAs-type disordered structure illustrated in Fig. 1C. There is no appreciable difference between the simulated XRD profiles of the CdI 2 -type ordered structure and NiAstype disordered structure. The experimental XRD profile shown in Fig. S2A mostly fit the simulated profiles for both the stoichiometric CdI 2 -type ordered and NiAs-type disordered structures. This suggests that it is difficult to evaluate the degree of ordering from the XRD analysis alone. However, Sokolovich and Bayukov concluded from Mössbauer spectroscopy experiment that similar but stoichiometric (CrFe)S specimens synthesized by sintering and quenching have a NiAs-type disordered structure rather than a CdI 2 -type ordered structure (2).
The chemical composition for the sample obtained by ICP-AES measurements was 22.7% Cr-23.3% Fe-54.0% S (at.%). This is nearly identical to the nominal composition of 23% Cr-23% Fe-54% S (at.%). Since the (CrFe)S sample synthesized in this study has a S-rich off-stoichiometric composition with an Fe/Cr ratio of approximately 1, the excess S atoms and/or vacancies (Vc) are expected to be distributed in the 2a sites to retain a single-phase NiAs-type structure. Figs. S2D and E present offstoichiometric NiAs-type structures with the composition (Cr 23 Fe 23 )S 54 . Fig. S2D is obtained by assuming that the excess S atoms (4 at.%) randomly substitute for Cr/Fe at the 2a sites (i.e., the sample can be represented as (Cr 23 Fe 23 S 4 )S 50 ), while  Fig. S2E, rather than the simulated patterns for the (Cr 23 Fe 23 S 4 )S 50 lattice shown in Fig. S2D. This indicates that the S-rich (CrFe)S compound synthesized in this study potentially has a NiAs-type disordered structure containing a certain number of vacancies in the 2a sites. It may be noted that 7.4% of vacancies in the 2a sites might be excessive for retaining the crystal structure and electronic state of the NiAs-type lattice. Therefore, it may be reasonably assumed that the 2a sites in the experimental samples are occupied not only by the vacancies, but also by a portion of excess S atoms along with Cr and Fe atoms. To elucidate the detailed structure and degree of ordering of the (CrFe)S compound synthesized, neutron diffraction analysis is ultimately required, which will be reported in the future.
Temperature dependence of the magnetization Behavior of the M-T curevs of ferrimagnets differs depending on the strength of the exchange interaction of the sublattices, as predicted by Néel (3,4), and categorised into several types. Fig. S3 indicates the examples of the temperature dependences of the magnetization of each sub-lattice (red and blue) and the total magnetization (green), which is simply the sum of the magnetizations of the two sublattices. The R, N, and P-type behaviours have been already predicted by Néel. The right-bottom figure shows the possibility of the upward convex beahviour observed just below the Curie temperature (called here as NP-type), in which the temperature dependence of magnetization of each sub-lattice varies from N to P-type. Thus, the upward convex behavior can be seen as a result of the mixture of different behaviours in the sublattice magnetizations.
Theoretical calculation of the density of states It is important for first-principles calculations to use proper crystallographic parameters such as lattice parameters and atomic arrangement in the unit cell. The lattice parameters of stoichiometric NiAs-type (CrFe)S are predicted to be a = 0.3456 nm and c = 0.5875 nm based on first-principles calculations. The DOS is also drawn, as shown in Fig. 4. The lattice parameters of the (CrFe)S sample in this study, however, were measured to be a = 0.3446 nm and c = 0.5743 nm based on the XRD profile shown in Fig. S2A with an expected experimental accuracy of ±0.0010 nm. If we performed the first-principles calculations using the experimental lattice parameters, we obtained the DOS shown in Fig. S4 for the stoichiometric CdI 2 -type ordered and NiAs-type disordered (CrFe)S compounds, as well as the values of physical properties such as magnetic moment and Curie temperature. We find that these are not significantly different from the theoretical values obtained by first-principles calculations, presented in Fig. 3 and Table 1. This indicates that the (CrFe)S compounds exhibits a half-metallicity even if the lattice parameters fluctuate slightly.
Since the (CrFe)S sample synthesized in this study has an off-stoichiometric composition, the atomic arrangement cannot be strictly determined from the XRD profile as discussed before. We assume that the 2a sites in the NiAs-type structure are randomly occupied by Cr, Fe, excessive S atoms, and vacancies (Vc). For example, if the structure assumed represented as (Cr 21.8 Fe 21.8 S 1.4 Vc 5.0 )S 50 , the DOS becomes one shown in Fig. S5. We can see that the DOS shown in Fig. S5 is essentially the same as the DOS shown in Figs. 5 and S4. Thus, it is concluded that the half-metal-type electronic state is retained as long as the total number of d-electrons per magnetic ion is 5 (when the Fe/Cr ratio of the (CrFe)S compound is 1 in the case of the NiAs-type (CrFe)S compound), even if some vacancies exist and the S content deviates from the stoichiometry. We thus conclude that (CrFe)S compounds with S content ranging from 50 to 55 at.%, including the (Cr 23 Fe 23 )S 54 compound synthesized in this study exhibit halfmetallic fully compensated ferrimagnetism.    S3. Examples of the temperature dependencies of the magnetization of each sub-lattice (red and blue) and the total magnetization (green) that is simply obtained as a sum of the magnetizations of the two sub-lattices. R, N, and P-type behavior have been predicted by Néel (2,3). The right-bottom figure (NP-type) indicates a possibility of the upward convex beahviour, observed experimentally, just below the Curie temperature, where the temperature dependence of magnetization of each sub-lattice varies from N to P-type.